Questions tagged [billiards]

For questions about billards, a traditional tabletop game played with balls, sticks called cues and a specialized table.

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In billiard systems, why are birkhoff coordinates needed to create area preserving maps?

Birkhoff co-ordinates, when used to obtain Poincaré sections of a billiards dynamics are often referred to as 'area preserving'.. why ?
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Verifying that a billiard system satisfies the twist condition.

I'm numerically studying a 2D billiard system whose domain is a unit outer circle with an inner elliptical scatterer of variable geometry. EDIT: Both the circle and ellipse share a common centre. ...
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Why are these angles equal?

-see picture above- (from Billiards and Geometry by Serge Tabachnikov) I don't understand why the angles $F_2BA_1$ and $F_1BA_0$ are equal (I do understand the conclusion, that follows from the ...
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Types of triangles admitting periodic billiard orbits

It is an open problem in dynamical systems if every triangle has a periodic billiard orbit. So far it has been proven that equilateral triangles, isosceles triangles, right triangles, and triangles ...
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Billiards in a holey square

Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
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52 views

Chaotic system?

I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region. Where I need help in understanding how I might show ...
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Joining boundary points through affine/billiards trajectories

Assume we have a smooth bounded domain $\Omega \subset \mathbb{R}^d$, and two points at the boundary $x, y$. I want to show that there exists an integer $n$ such that one can draw an affine path ...
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Billiard ball mental patients.

The Question: Suppose there are $n$ extremely paranoid, vulnerable mental patients at a hospital. Each day at the lunch hour, they move around like frictionless billiard balls of radius $\rho$...
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Unfolding a billiard trajectory on an unbounded billiard table

A common "trick" used in the study of polygonal billiards is to unfold the trajectory. I'm interested in billiards which may be bounded in one direction, and unbounded in another. For instance, ...
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37 views

Reflecting a vector within a box (variation on the billiards problem)

Let there be a box, with bottom left corner at (0,0), and top right corner being at (m,n), where m and n are positive integers. A starting point is chosen at random within the box at (x,y), such ...
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The simplest billiards problem

The Problem: Let's say we have a rectangle of size $m \times n$ centered at the origin (or, if it makes the math easier, you can place it wherever on the plane). We take a billiard ball, ...
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Simple Billiard Table with Positive Lyapunov exponent

I can understand the Lyapunov exponent for a circular billiard table is zero. However, if I were right, I read the Lyapunov exponent for a stadium billiard table is non-zero which was surprising for ...
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Trajectories in circular Billards

Given two points $p_1,p_2$ on a circular billard table. I want to know all billard trajectories from $p_1$ to $p_2$ hitting the boundary precisely once. Model of the circular billard: Denote the ...
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Natural projection of the billiard phase space

Setup The phase space of the billiard flow $\left\lbrace \Phi^t \right\rbrace$ ($t \in \mathbb{R}$) is given by $\Omega = \mathcal{D} \times S^1$ where $\mathcal{D}$ is a planar billiard table in $\...
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About a theorem by Mather on smooth convex billiards

I'm a masters student and in about a week I begin working in a short presentation of "a theorem" by John N. Mather. More precisely, I have to give some elements of the proof of a theorem that states ...
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Infinitely many triangular numbers which are of form $n^2-1$

After playing a round of pool billiard recently, I noticed that the fifteen colored balls can be arranged in a triangle (obviously) and all sixteen balls (including the white ball) can be arranged in ...
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Launching billiard balls at 45 degree angles and bouncing of off edges.

Say I have a billiard ball and I launch it from the bottom-left corner of a table with length $x$ and width $y$. Given $x$ and $y$ will the ball reach a corner again, which corner and in how many ...
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Number of possible rays in rectangular mirror room between two points

Question There is a rectangular room with mirrors of $a$ width and $b$ height (Aligned with co-ordinate axes, so lower left corner is at $(0, 0)$ and upper right corner is at $(a, b)$). There are two ...
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Quantitative estimates on space filling curves

To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed ...
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288 views

Tangent to hyperbola meeting on ellipse

Can anyone prove that the two tangent lines to hyperbola that meet on a confocal ellipse make the same angle with the tangent of the ellipse at the point of intersection? Thanks
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Area of a billiard and its preservation

From S. Tabachnikov's Geometry and billiards Consider a plane billiard table $D$ whose boundary is a smooth closed curve $γ$. Let $M$ be the space of unit tangent vectors $(x, v)$ whose foot ...
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225 views

About trajectories on a circular billiard

Let consider a circular billiard. You start by putting the ball on the edge off the table. The trajectory then is quite simple (see the picture), with equal angles at each rebound. Which will ...
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Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
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Billiards in a circular table

This is a variation of Alhazen's Billiard Problem. Suppose we have a semicircular billiards table of radius r centered at the origin O, and a billiard ball placed somewhere on the 'x-axis' of the ...
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Trajectories on a circular billiards table

This question is related to The case of Captain America's shield: a variation of Alhazen's Billard problem, but more focused. Let the unit disc in the plane be our billiards table, and let $C$ ...
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Is there an elementary proof for the envelope of an infinitely-bouncing billiard ball?

I was reading this page here http://cage.ugent.be/~hs/billiards/billiards.html And was wondering if there existed an elementary proof of proving that the envelopes form certain conic sections (ie: ...
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For points A, B, does there there a billiard such that any trajectory from A will reflect twice and then reach B?

I'm looking for a kind of generalisation of an ellipse; a shape with a more complicated optical property. I'm not sure how to rigorously define this shape, or prove that it exists, or find an equation ...
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52 views

Is it true that these angles are equal?

Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to ...
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Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
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Constants of motion for non-chaotric billiards

As I understand it, for a mechanical system, each symmetry leads to a constant of motion. For integrable systems, the number of constants of motion equals the number of degrees of freedom. So take 2D ...
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Turning a rhombus billiard into an equivalent barrier billiard

I am reading a research paper and the authors map the rhombus billiard (angles $60$-$120$ degrees) to an equivalent barrier billiard. They start with a rhombus standing upright and reflect in a ...
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Counting Periodic Orbits on a regular Hexagon

An orbit on a polygon is a path that a "billiards ball" (a point) would follow if it obeyed Snell's law of reflection (the angle of incidence is equal to the angle of reflection). A periodic orbit is ...
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Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
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Tool for generating mathematical billiards

i'm currently looking for some program which can generate arbitrary mathematical billiards and trajectories inside them. Any help would be much appreciated! Thanks/ Mats
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Perfectly centered break of a perfectly aligned pool ball rack

This question is asked on Physics SE and MathOverflow by somebody else. I don't think it belongs there, but rather here (for reasons given there in my comments there; edit: now self-removed). ...
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The forever moving billiard ball

Suppose I have a rectangular table, dimensions $x$ by $y$, and a billiard ball is positioned in the very center. For descriptive convenience, let us impose a coordinate system on this table with an ...
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Polygonal billiards and uniform distribution

According to this article in Wikipedia: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle ...